Cyclicity of bicyclic operators and completeness of translates

Evgeny Abakumov, Aharon Atzmon, Sophie Grivaux*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if S: (an) nεℤ→ an-1nεℤ is the shift operator acting on the weighted space of sequences ℓ2ℤ , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if limn→ +rm log}ω(-n){n}=0} . On the other hand one can see that S is not cyclic if the series ∑ngeq 1} logω (-n)}/n2 diverges. We show that the question of Herrero whether either S or S* is cyclic on ℓω2 ℤ admits a positive answer when the series ∑nεℤ rm log} S{n}||/(n2+1)} is convergent. We also prove completeness results for translates in certain Banach spaces of functions on ℝ.

Original languageEnglish
Pages (from-to)293-322
Number of pages30
JournalMathematische Annalen
Issue number2
StatePublished - Jun 2008


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